Hi Andrw,
Unweighted and weighted UniFrac are providing information on two different things. Unweighted takes into account presence/absence, while weighted takes into account abundance also. This certainly can yield different p-values.
Bray Curtis is a non-phylogenetic metric that like weighted UniFrac takes into account abundance. However, because Bray Curtis is a non-phylogenetic metric and weighted UniFrac is a phylogenetic metric that takes into account branch length on the phylogenetic tree, very small branch lengths on the tree might explain why something that is significant with Bray Curtis is insignificant with weighted UniFrac.
Thanks!
Embriette